Consolidation of Axiomatic Set Theory

Abstract

After 1918, most important contributions to the foundations of abstract set theory relied on modern axiom systems. But until the 1920s few authors adopted Zermelo’s axiom system explicitly.2 As we saw in the preceding chapter, many favored the theory of types because it seemed to offer a safer framework, and at the same time it was sufficient for the limited amount of set theory that is necessary in so-called classical mathematics. As late as 1939 Alonzo Church was writing that the simplified theory of types and Zermelo’s set theory were essentially similar, and the “safest cities of refuge” for classicist mathematicians at the time.3 But the Zermelo system had to compete with another alternative, the system of von Neumann, presented in 1925 and developed later by Bernays and Gödel (see §3).